You are conflating Bayes' Theorem with Bayesian Inference generally. Bayes Theorem is, by itself, insufficient to provide any of the important things - like a confidence interval. The Bay Area rationalists tell themselves "we can just plug the sample into Bayes' Theorem, and we're done." That is NOT modern methodology, and Dirichlet was only an example of modern techniques providing superior information. That's why I mentioned SAS, as well. Bayesian inference is correct, and I am a Bayesian not a Frequentist. AND - if you're just plugging numbers into P(A/B) without doing all the other work first, you're definitely wrong. That rationalist community's luminaries make exactly that mistake.